U. S. Department of Commerce
National Bureau of Standards
Research Paper RP2059
Volume 44, January 1950
Part of the Journal of Research of the National Bureau of Standards
Solution of the Telegraphers Equation With Boundary
Conditions on Only One Characteristic ]
By George E. Forsythe
Forecasting a certain idealized horizontal, autobarotropic, nonviscous, nondiverging
atmospheric flow considered by Rossby leads to an unusual boundary-value problem for
the telegrapher's equation, involving boundary values on only one characteristic. It is
shown how to find unique solutions periodic in the longitude; these are represented in terms
of a Green's function, A procedure for computing the Green's function is set down and is
shown to be optimal in a restricted sense. The Green's function is tabulated for 72 longi-
tudes and 14 time-values. An alternative solution by a difference equation is mentioned.
I. Introduction
In one treatment of planetary atmospheric flow
as horizontal, autobarotropic, nonviscous, and
nondiverging in a plane, Rossby [9] 2 considered
the idealized case of a constant west-wind com-
ponent, U, and a south-wind component, v, depend-
ent on the west-to-east distance coordinate,^, and
time, r, but independent of the south-to-north
distance coordinate, /x. It was shown in [5] that this
v satisfies the telegrapher's equation (eq 3) below,
where x=\[/—Ut, f=4 fir. The parameter £=
2Q cos <p (d<p /did) is here considered constant; 12 is the
angular speed of the earth's rotation, and <p is
latitude. For this simple atmospheric model, the
meteorological forecast problem is one of deter-
mining v(x, t) for future times t, given only v(x f 0).
But on a plane the specification of v(x, 0) is not
sufficient to determine v(x, t) for many £>0, be-
cause the line t—0 is a characteristic of eq 3 (see
p. 254 of [11]). Having its initial conditions on
only one characteristic is an unusual feature of
the present problem that does not seem to have
arisen in other physical problems known to the
author to lead to the telegrapher's equation.
The author shows that the forecast problem has
a unique solution when it is assumed that the world
is round, that is, when the solution is assumed to
1 This paper was written at the Institute for Numerical Analysis of the
National Bureau of Standards with the financial support of the Oflice of
Naval Research of the U. S. Navy Department.
2 Figures in brackets indicate the literature references at the end of this
paper.
be periodic in x. The problem is stated in section
II and solved in section III. In section IV the
solution is represented in terms of a Green's func-
tion. In section V a procedure is outlined for
computing the Green's function by improving the
convergence of its Fourier series. In section VI
certain auxiliary polynomials, a k (x), used in
seel ion V are discussed and related to the Bernoulli
polynomials. In section VII are reported without
proof a few results on the approximate solution of
the problem by a difference equation, taken from
[(>]. In section VIII is given a table of values of
the Green's function, as computed in the Compu-
tation Unit of the Institute for Numerical Analysis.
The present author first reported this work in
[7]. Independently of the research reported here,
Charney, Eliassen, and Hunt of the Institute for
Advanced Study considered the telegrapher's
equation while investigating numerical weather
prediction in general. Their research was reported
in [1] and is written up in [2]. The work of these
men includes much of what is reported here, and
much more.
II. Statement of the Problem
Let C be the circumference of a unit circle; let
us adopt an angle coordinate x for C\ — tt<^x<tt.
Let / be the set of time-instants t: 0<t<^ °° . Let
R be the closed two-dimensional region consisting
of all points (x, t) with x in C and t in /. Let f(x)
be a real-valued function that satisfies the follow-
Solution of the Telegrapher's Equation
89
ing hypotheses, but which is otherwise arbitrary:
Hi. f(x) is sectionally smooth 3 on C. Moreover,
f(x)=i[f(x+0)+f(x-0)], (all x).
H 2 : fix) has the average value zero:
f(r)dx=0.
/:
(i)
(2)
The problem is to find a real-valued function,
v(x, t), defined everywhere on i?,with the following
four properties:
Pi'. v t exists* and is continuous throughout R.
P 2 : v x and v tx =c)Vt/c>x exist and are continuous
everywhere in R except, at most, for a finite number
of values of 5 x.
Pz'. Whenever v xt is defined, the following hyper-
bolic partial differential equation (the telegrapher's
equation) is satisfied:
v xt +lv=0. (3)
P 4 : For t=0, v(x, t) reduces tof(x):
»(3,0)=/(3).
III. Solution of the Problem, Uniqueness
One gets a formal solution by separation of
variables 'and use of Fourier series. Assume a
solution of eq 3 of form v(x, t) =X(x) T{t) . Then
v x .=X'(x)T'(t), and eq 3 takes the form
X'V) T'{t)
X(x) ' T(t) -
(4)
The two factors in eq 4 must themselves be
constant:
X'(x) ,
X(x)
T'(t) _
T(t)
_1_
"4\*
(5)
(6)
A solution of eq 5 for — oo<^x<oo is X(x)=e Kx .
For x on the circle (7, however, one must have
X(—Tr)=X(ir), or e"* x =€* x . Taking logarithms,
oneseesthat — 7rX=7rX+2n^(?i=0, ±1, ±2, . . .).
Hence X=m(7i=0, ±1, ±2, . . .). Since the value
3 That is, both f(x) and/'(z) are continuous in C except for a finite number
of jump discontinuities.
4 The subscripts denote partial derivatives.
» It is shown on pp. 55 to 57 of [3] that our conditions Pi and P 2 imply the
following: r xt exists and equals Vt x for all (x, t) such that x is not one of the
excepted values in P 2 .
X=0 is incompatible with eq 4, there remain the
following fundamental solutions of eq 5:
X n (x)--
(n=±l,±2, ±3,
.)■
Corresponding to X n (x), the solution T n (t) of eq 6
for \=ni is T n (t) = exp (it/4n) . Hence for n= ± 1 ,
±2, . . . the functions X n (x)T n (f)=exp [i(nx+
t/4:n)] have properties P u P 2 , and P 3 . By taking
linear combinations of the functions X n T n and
X_ n T_ nj one obtains the equivalent pair of func-
tions cos (nx-\-t/4:n) and sin (nx-\-t/4ri). Both of
the latter functions have properties P l9 P 2 , and P 3 .
In order to obtain a solution with enough de-
grees of freedom to satisfy P 4 , consider the series,
v(x,f)'
-§ [ a » cos ( nx+ L) +b " sin ( w+ ^)]'
(7)
where a n , b n are undetermined constants. We
postpone a discussion of the convergence of the
series (eq 7) for t^O and consider it for t=0, where
v(x } 0) is supposed to equal f(x):
v(x,0) ^^Li(a n cos nx-\-b n smnx).
71 = 1
(8)
If the series in eq 8 actually does converge tof(x)
for all x, it is shown on p. 274 of [12] that the co-
efficients, a n , b n , must be the Fourier coefficients of
/. Conversely, by p. 25 of [12], the hypothesis
Hi is sufficient to insure that the Fourier series of
/ actually converges to f(x) ; it even converges uni-
formly for x in any interval bounded away from
a discontinuity of /. Moreover, the hypothesis
H 2 implies that in the Fourier series a =0. We
henceforth stipulate that the series (eq 8) is the
Fourier series of/. It is important to note 6 that
Hi implies that a n and b n are 0(l/n); that is, there
exists a constant M <C °° such that
\na n \<M, \nb n \<M (all n).
1. Proof of Convergence
(9)
There remains only a proof that the series in
eq 7 actually does converge to a function v(x, t)
with the required properties, Pi,P 2 , P 3 , P 4 . It will
be useful to have the following representations of
cos (tjAn) and sin (tjkn). They are proved by
Taylor's formula and hold for all values of t/An:
« See p. 18 of [12].
90
Journal of Research
cos-j— =1"
An
* n t 2
32n 2
where|a„|=a^j|<l;
. t U
An An
where | /S,| =|^(^) ! <1;
sin 4n 4n 384n 3 '
(10)
(11)
(12)
where \y n \ =|t(^)| <1.
Using eq 10 and 11, one sees that for any fixed t,
S «* cos(nx+^J+b n sm(nx+ 4 - ) I
S ( a n cos "£+&* sin nx) cos vz +
(&„ cos nx— a„ sin ma;) sin ~. =
oo
S ( a « cos nx+bn sin m) —
71 = 1
t^ (b n a n . \
-7> j p n ( — cos nx sin nx )=
4^ ^ n \n n J
(13)
Representation as the sum of three series is per-
mitted because each of the series 2 , S x , S 2
converges. 2 converges for all x because it is
the Fourier series of f ; its convergence is uniform
in any interval bounded away from a discontinuity
of f(x). Fix any positive number t l9 and restrict
the consideration to t'& such that 0<t<ti. Since
a n and b n are 0(l/n) 9 2i and 2 2 are convergent
uniformly in x and 2. For example, 2 2 is domi-
nated by (U/A) ^,(\b n \/n-\-\a n \/ri), a series con-
vergent like 2(lM 2 )- The series of eq 7 is thus
convergent for all x, t and defines by its limit a
function v(x, t):
v(x, 0=S &n cos (^+4^)+&n sin ( nx+~ ) •
(14)
Moreover, the series (eq 14) converges uniformly
for x, t such that x is bounded away from a jump
of/(x).
Since Si and 2 2 converge uniformly, they con-
verge to continuous functions of x and t. Thus
the only discontinuities of v{x, t) are those from
S , that is, those of j(x). This is the property of
hyperbolic differential equations that discon-
tinuities in their solutions are propagated along
characteristics. Let the set of discontinuities
of J(x) be denoted by E.
For all x, t one may obtain v t by termwise dif-
ferentiation of eq 14, because by eq 9 the resulting
series is absolutely and uniformly convergent in
both x and t:
*(*, =*£ g cos (W ±)-<% sin (««+£)}
(15)
Moreover, v t is continuous for all x, t, so that Pi
holds. Now one may not obtain v x by termwise
differentiation of eq 14, because the resulting
series will generally not converge. However,
v x does exist and is a continuous function of x
and t for all t and for all x not in E. To see this,
one uses eq 10 and 12 to carry the Taylor formula
(eq 13) to one higher power of t. It is found that
)(Xj t) =f(x) +t ^ ( "
'K
n
On
n
)-
t 2
32
b n
cos nx-\ — 2 sni 7i ^
■)-
3kS 7 »(«» COS7w: -S 8in7la; } (16)
By termwise differentiation of eq 16, it is found
that
4- OO
#*(#, t) =j r (jJ-tS (a B cos nx+ 6 n sin nx) —
4 71 = 1
t 2 ™ /b n a n . \ .
™ >J a w ( — cos nx sin nx ) +
32 ^i n \n n )
384
s*&
cos tmH — 2 sin nx
/'(x)-i#)+2 3 +2,
Restricting attention to t with ^ ^ /i< °° , one
sees that the expressions 2 3 and S 4 are uniformly
convergent with respect to x and t. The series
leading to j(x) is uniformly convergent for x
in any interval bounded away from a discontinuity
Solution of the Telegrapher's Equation
91
of /(a?). Hence, for x not in E, termwise differ-
entiation leads to the correct value of v x . More-
over, v x is continuous in x and t whenever a; is a
point of continuity of f(x). To get v tx one may
differentiate eq 15 termwise with respect to x:
v tx (x,t) =
"*S[ a - co *( nx+ in) +bn sin (^+^)} < 17 )
As remarked after eq 14, the series in eq 17 is
uniformly convergent for x, t such that x is bounded
away from the discontinuities of f(x). Hence
v tx is continuous in x and t for all x, t except for
x in E. Since v t , v x , v tx are all continuous, v xt
exists and equals v tx except on the lines corre-
sponding to the discontinuities of /(a?). This
shows that v(x, t) has property P 2 . The eq 17
and 14 show that v satisfies the telegrapher's
equation (eq 3). Finally, property P 4 was taken
care of by the selection of {a ny b n }. Thus the
problem is solved completely.
2. Proof of Uniqueness
It will be shown that v(x } t) is the only function
that solves the above problem. Suppose that
Vi(x,i) were a second solution. Then the differ-
ence, w(x,t)=v—Vi, satisfies the same problem with
f(x) =0. For each x u by property Pi, w{x u t) and
w t (xij t) are continuous functions of t for 0^t<C °°,
while w(x, 0) = and w x {x, 0)^=0 are, of course,
continuous functions of x. " ' For each t, let the
value of w(x, t) be extended as a periodic function
of x to all x in the interval [— 2ir, 2ir]. Now the
values w{— ir, t) and w(x, 0) are given on two charac-
teristics of eq 3. By pp. 21 to 22 of [10] they are
therefore sufficient to determine w(x, t) uniquely
for all x, t. On the other hand, the values w(t, t)
and w(x, 0) are also sufficient to determine w(x,t)
for all x, t. Since w(—tt, t) =w (vr, i) and w(x, 0) =
w(— x, 0) = 0, it is seen by symmetry that w(x, t) =
w(—x,t). Now since the values of x lie on a circle,
there is nothing exceptional about the line£=7r.
The above argument will also show that, for each
value of Xi, w{x\-\-x } t) = w(xi—x, t). It follows that
for each fixed t, w(x, t) = constant, whence w(x, t) =
hit). Byeq3, —iw=w tx z= (d/dx)h / (t) = 0. Hence,
the constant value of w(x, t) must be everywhere
zero. Then v=v h and the solution v(x, t) given by
eq 14 is unique.
The results of section III may be summarized
in the following theorem, phrased in the notation
of section II.
Theorem 1. If the real-valued junction f(x)
defined on C satisfies hypotheses Hi and H 2 , then
there exists a unique function, v(x, t), defined on
R and possessing properties P u P 2 , Pz, and P^
If eq 8 is the Fourier series of f(x), then v(x, t) is
defined explicitly by eq 1/+.
It is of mathematical interest 7 to note that
Theorem 1 can be extended to general functions
f(x) of bounded variation. That is, one may re-
place Hi by the weaker hypothesis
Hi: f(x) is of bounded variation on C. Moreover*
f(x)=i[f(x+0)+f(x-0)], (all*).
The solution v(x, t) is required to have property
P 2 , instead of P 2 :
P 2 : v x exists in R except for x in a set E (EcC)
of Lebesgue measure zero; for all t and for all x not
in E, v tx exists and is a continuous function of x and d t.
The extension of Theorem 1 is stated as follows:
Theorem 2. If the real-valued function f(x)
defined on C satisfies hypotheses Hi and H' 2 , then
there exists a unique function v(x, t) defined on R
and possessing properties P u P' 2 , P 3 , and P 4 . If
eq 8 is the Fourier series of f(x), then v(x, t) is
defined explicitly by eq H.
The convergence proof of section III, 2 requires
only slight modification to serve as a proof of
Theorem 2. For an arbitrary function f(x) of
bounded variation, there need be no interval of
continuity; one may therefore not expect the
series (eq 14) to converge uniformly in any
interval. The termwise differentiations of section
III, 2 can, however, be justified for almost all x by
the fact that the resulting series are Fourier series.
IV. Representation by a Green's Function
The formula (eq 14) for the solution of the
problem of section II is directly adapted to
numerical computation only when the Fourier
coefficients a n) b n converge rapidly to zero. But
some of the most important cases in meteorology
are where f{x) has discontinuities (see footnote 7).
7 This extension seems to have no meteorological interest. However, it is
of much importance in meteorology to deal with functions f{x) with some dis-
continuities; such discontinuities occur at fronts between air masses.
8 Same aseq 1.
9 It follows from Pi and P' 2 that v xt exists and equals v tx for all t and for all
x not in E; see pp. 55 to 57 of [3].
92
Journal of Research
With such an / the Fourier coefficients are,
roughly speaking, of the order 0(1 /n), and for
those / the convergence of eq 14 is hopelessly
slow.
It is possible, however, to improve the con-
vergence of eq 14 to such a degree that computa-
tion of v(x y t) is reasonably possible. The pro-
cedure will be illustrated in section V for one
particular choice of f(x) :
v sin nx
f(x)=<r (x)--
71-1
(0<x<x)
(x=0)
■iU+x) (-w<x<0),
(18)
It may be shown by direct computation that the
Fourier series of oo(x) is the series of eq 18. It
then follows that the series converges to a {) (x) for
all ./'. The reason for choosing a (x) is two-fold:
(a) it is of meteorological interest to see how a
simple discontinuity in v(x, 0) is propagated, as t
increases; (b) for any f(x) that is sectionally
smooth, it is possible to represent the correspond-
ing v(x, t) in terms of the solution for the special
initial condition v(x, Q)=a (x).
The present section is devoted to proving the
property (b). Suppose, therefore, that G(x,t) is
the solution of the problem of section II with the
initial condition <r (x)) then G(x, 0) = a (x). Let
a sectionally smooth function f(x) be given that
satisfies eq 1 and 2. Let /(#) have the jump J k =
f(x k +0)—f(x k —0) at the point x k (k = l, 2, . . . ,K).
Let Co Or) be continued periodically for x in [— 2ir,
2w\. Then (J k /ir)a (x—x k ) also has the jump J k
at the point x k . Now
1 K
£0*0 =f( x ) S Jk<r (x—x k )
7Tfc=i
is a continuous function, since all the jumps have
been removed. Moreover, £(x) and the functions
(J k /ir)<r (x—x k ) all satisfy eq 1 and 2. There is,
therefore, a unique solution to the problem of
section II for each of these functions. For the
function (J k /Tr)<r (x—x k ) the solution is (Jjc/tt)
G(x—x k , t). It will be shown below that the solu-
tion y(x, t) corresponding to the initial values £(x)
is given by
1 f *
y(x,t)=- G(x-u,t)£'(u)du. (19)
Since the problem of section II is linear in the
initial condition /(#) , and since
/(*) =*(*)+- S eWs-z*), (20)
it follows that
1 K I r^
v ( Xj t)=-^J k G(x—x k ,t)+- G(x-u,t)Z'(u)du.
(21)
The formula (eq 21) is the desired representation
of v(x, t) in terms of the solution G(x, t) to the single
problem where f(x) = o- (a;) . The nature of eq 21
indicates that G(x,t) may be called the Green 7 s
function of the problem of section II. The repre-
sentation (eq 21) is not only of theoretical impor-
tance, but it can also be used for approximating
the solutions for general boundary values, fix),
once the Green's function is tabulated. The prac-
tical problem then becomes one of approximating
the integral in eq 21 by some numerical process.
This latter problem is not treated here.
It remains to prove eq 19. First, it may be
observed that, for each x, since <r and £ are
periodic,
a (x—u)£' (u)du=- a (;x—u)d£(u) =
Wj-TT TTj-TT
i r *■ i r x ~° ,
£('u)d u <T (x — u) =- £(u)a (x—u)d
TTJ-tt TTJ-tt
II
i C w
+ t(x)+- £(u)a (x—u)du=
irjx+o
1 f*
£ (x) - ^ J _J (u) du=£ (x) .
(22)
In eq 22 we have used two Riemann-Stieltjes in-
tegrals. The last step is true because £(x)
satisfies eq 2. By eq 13,
G(x, f)=a Q (x) +J S § cos nx-~ J2 °^ sin nx.
Hence
if* i f 7r t f 71 " r^% "l
G(x—u,t)£'(u)du = ~ <r (x—u)£'(u)du-{--7— z_, ~| (cos nx cos nu-{- sin nx sin nu) \£'(u)du—
7T J-7T KJ-Tr 47T J-ttLw=1 n J
t 2 f* r " a n/ . . ."!,.,. . . (23)
^T" > i -s (sm nx cos nu— cos nx sin nu) \£(u)du.
6ZT J-7T L W=l n J
Solution of the Telegrapher's Equation
93
Since £(u) is sectionally smooth, \£'(u)\ is bounded.
Hence the series in eq 23 remain uniformly con-
vergent when multiplied by %'(u) and may be inte-
grated termwise. We note that
- cos nu%'(u)du=- sin nu%(u)du= nb n ,
7T J-7T 7T J-7T
- $mnu£'(u)du= : cos nu£(u)du= — n a n ,
(24)
where a n and b n are the Fourier coefficients of f.
In view of eq 22 and 24, the termwise integra-
tion in eq 23 yields
- C G(x-u,t)Z'(u)du=ttx) +
TTJ-TT
t °° B
j^Lj— (b n cos nx—a n sin nx) —
4 n= i n
OL n
™ 2 d? («n cos w«+ ft« sin nx) .
oz w= ] n
(25)
But, by eq 13 and 14, the right-hand side of eq 25
is y(x,f), the solution corresponding to the initial
values £(#). This completes the proof of eq 19.
The representations (eq 20 and 21) assume a
more symmetric and unified form when the Le-
besgue-Stieltjes integral is used. It can be shown
that
/(*)=-( <T (x-u)df(u), (26)
and that
»(M)=-f G(x-u,t)df(u), (27)
TTJ C
where eq 26 and 27 include Lebesgue-Stieltjes in-
tegrals over the circle C. Whenever f(x) has a
discontinuity (say at x{) , the integral in eq 26 fails
to converge as a Kiemann-Stieltjes integral for
x=Xi, because the. functions o- (xi—u) and f(u)
both have a discontinuity for u=0. The same
holds for eq 27. The integrals (eq 26 and 27) are
convergent for all x as Lebesgue-Stieltjes integrals.
Moreover, the formula (eq 27) yields the solution of
the problem when f(x) is an arbitrary function of
bounded variation; the above proof of eq 19 can
be modified to serve as a proof of eq 27.
V. Computation of the Green's Function
For the purpose of using the representation (eq
21) and for its own meteorological interest, it was
desired to compute the Green's function G(x, t).
A tabulation to three decimal places, accurate to
approximately 0.001, appeared sufficiently accu-
rate. An x interval of 5 degrees of longitude
(7r/36 radian) is convenient in meteorology. It
was decided to compute G(x, t) for various times
t up to 96 hours at latitudes <f> from32°19 / to 10
55°. Since the length unit is here the radian of
longitude, the expression d<t>/djjL in section I takes
the value cos <j>. Then 40=812 cos 2 <£, where
ft=7.292X10" 5 radian/second. Now 24 hours
corresponds to r =86,400 seconds, or to t= 50.40
cos 2 <£. At latitude 32°19 / , cos 2 0=0.714, whence
£=36 at 24 hours. The largest value of t for
which G(x, t) was computed corresponds to 96
hours at latitude 32°19'; it is £=144.
Let 2=//4, for convenience. By eq 14
G(x, z) =X1 - sin ( nx-\ — )•
(28)
In summary, a method is required to compute
G(x, z) to an accuracy of approximately ±0.001
for 11 x= — 7r(7r/36)7r and for various positive values
of z up to 36. The present section will present
one such procedure, an application of a
method for improving the convergence of certain
Fourier series, given on pp. 84 to 88 of [10]. The
'procedure presented below is not an exact description
of the methods actually used in making the table of
section VIII. It is assumed in section V that
computing machinery is available capable of
dealing with numbers of 10 decimal digits, but
no more than 10.
Of the tolerable error 0.001, the amount 0.0005
must be reserved for round-off in the final tabula-
tion to three decimal places. Suppose that 0.0004
is allowed for truncation errors, 12 and 0.0001 for
computing errors resulting from round-off s during
the calculation with 10-digit numbers. To have a
truncation error as low as 0.0004 from use of a
partial sum of eq 28 would require about 23,000
!« The limit <£=32°19' arose unintentionally.
11 The notation x=a(8)b means x=a, a-\-8, a+25, a+38, . . . , 6—5, b.
12 Truncation errors are errors that result from use of approximate mathe-
matical formulas, e. g., use of partial sums of infinite series.
94
Journal of Research
terms for x=7r/36 and #=357r/36. The con-
vergence must obviously be improved.
1. Representation by Truncated Double Sum
We write
00 1 z
G(x, z) = ^Pi — sin nx cos -+
n — l Jl 71
00 I 2
Xj - cos nz sin -=Si+S 2 ; (29)
n = l ^ ^
for simplicity we consider only the first sum Si in
eq 29. It can be shown that the term S 2 behaves
similarly throughout the analysis. Expanding
cos (z/ri) in its Maclaurin series, one has
v-^ 1 • ^ ( -l) r z 2r _^ ^ (-l) r z 2r sin nx
Zl ~ir{ n sm nx fa (2r) !n» -£j fa (2r) to 2 '* 1
(30)
One type of truncation of eq 30 consists in
omitting all terms for n>N-\-l, r>R. Let the
error caused by this truncation be called ei. We
shall estimate e x for 0<z<Z:
«i =
n = A 7 +l r=/t!
(— l) r ^ 2r sin ra
(2r)!^ 2r+1
'! Jn x
« /eZV
oo V2r oo 1 oo 72r
= s
L'r i 1
£4 (2r) (2r)! ^ 4 V*
The last step above uses the Stirling expression
for the factorial function, which, according t o p.
74 of [8], is a one-sided estimate: s\ > s s e~ s ^2irs.
Continuing, one finds for <?Z/2i?iV<l that
i i^l f eZV R 1 ^( eZ V r
{eil< -±^-\2RNj B*>*fa\2RNj
4V7ri? 3/2
The first estimate in eq 31 seems crude, but it
does not affect the values of N or R very much.
Thus F(A T ,R) is an upper bound for the trunca-
tion error |ei| introduced by omitting terms of
type n>N+l, r>R in S lt
One can reasonably tolerate a truncation error
|ei| of 5X10 -5 . The corresponding admissible
values of N and R come from setting F(N,R) =
5X10 -5 for Z=36. The following pairs of values
of N,R were obtained from eq 31 by a numerical
calculation followed by a round-off of N to an
integral value:
N=2Q
18
13
11
iJ=4
N=9
7
4
R=S
10
15
20
(32)
The selection of the most suitable pair of values
N,R from eq 32 will be postponed until we have
discussed the summation of the remaining terms
of eq 30.
2. Computation of the Double Sum
The terms of Si for n=l, 2, . . . , N and all r
may be left in the form
S 1 =>,( - cos- )si
»=I \n n/
sm nx,
(33)
and may be computed from this formula. The
terms for n>N+l and r=0, 1, . . . , R—l may be
written in the form
where
S '~~§(2r)! a * )(x) '
4fr» = (-iy £^r
(34)
(35)
Once g^ (x) has been tabulated for the one value
of N to be selected below, Sg may be computed
directly from eq 34. Two methods are needed to
get a^ } (x), as the calculating machinery is as-
sumed to be limited to ten decimal digits.
The first method is to use the identity
c%\x) = c 2T {x)-{-iyJt S -^g' (36)
where
«2r(x)=^(x) = (-iy± S -^- (37)
In section VI it will be shown that for 0<x<7r,
g 2t {x) is essentially a Bernoulli polynomial in the
variable x/2w. Hence <r 2 r(x) can readily be corn-
Solution of the Telegrapher's Equation
862453—50-
95
puted or even interpolated from existing tables
like [4]. One can then get cr^ix) by carrying
out the subtraction indicated in eq 36. The terms
a 2r (x) and (sin x)/l of eq 36 are approximately
unity; with ten-digit calculating machinery these
terms may be carried to ten decimal places. Hence
0-2^ (x)j calculated from eq 36, will be good to
ten decimal places. In order to get the Green's
function to an accuracy of 0.001, it is necessary
that all terms in the sums of eq 33 and 34 be sub-
stantially correct to five decimal places; hence the
terms z 2r o^ (x) I (2r)\ in eq 34 must be given to five
decimal places. It follows that z 2r j(2r)\ must be
less than 10 5 ; for 2=36 this means that 2r<4.
Thus the first method of computing a^\x) is
adequate when 2r=0, 2, 4.
For 2r > 4, another way of getting o-^° (x) is
needed. For these larger r, the convergence of
the infinite series (eq 35) is good, and we may
write
<^>(^(-ir 2 S 4|?- (38)
n=N+l n
The truncation error e 2 in eq 38 may be shown
to be at its maximum when z=7r/36 and when
sin (N 2T x) =0. In this case the error is not greater
than the sum of the first 36 omitted terms, which
is estimated by
^ 36 s in (rnr/36) . 1
w=AT 2r +l
mr ^ 72
^iV 2 2 ; +1 tA ° ll± 3^7rN 2 2 ; +1
In order to keep € 2 z 2r /(2r)\ numerically less than
5X10 -5 for ^=36, it is sufficient that
KfJ +1 (^ =5Xl(r - (39)
Solution of eq 39 gives the following points of
truncation of eq 38:
iV 4 =126, N 5 =78, iV 6 =54, JV 7 =40,
^8=32,^=26,^0=22,....
(40)
(The values of N k for odd subscripts k are appro-
priate to a parallel analysis of the sum S 2 of eq 29,
but not to the present analysis of S x .) If ci N) (x),
<rs N) (x), . . . , oiftl.2 (x) are estimated by eq 38, with
the values N 2r taken from eq 40, the individual
summands of eq 34 will each have a truncation
error not exceeding 5X10 -5 .
3. Operational Analysis, Selection of N and R
We now estimate the labor involved in comput-
ing the Si of eq 29, in order to select that pair of
values of N and R from eq 32 that makes the
computational work a minimum. The resulting
computing procedure will be optimal in a limited
sense — i. e., optimal among the one-parameter
family of truncations considered in section V, 1.
Although the resulting procedure will be perfectly
feasible for computation — indeed, it differs only
moderately from the procedure actually used to
get the tables of section VIII — it cannot be said
to be optimal among all procedures for computing
the Si of eq 29. For it has been based on a certain
type of truncation of a certain double series (eq
30), and on a predetermined assignment of
truncation errors to several subcalculations (eq
38). Given only the nature of the computational
machinery, to describe an absolutely optimal
procedure of getting Si would seem quite beyond
the present powers of analysis.
It is customary and quite realistic to estimate
the cost of a computation by the number of
multiplications required. 13 We shall consider
the multiplications required to get Si for one
value of z and for one value of x. In eq 33 one
may ignore the desk computation required to get
(l/ri)cos(z/n) and sinnx; there are then essentially
N multiplications involved in eq 33. In using
eq 36, one may ignore the work of getting <j 2r (x),
which is chargeable to basic table development,
and count SN multiplications needed in all to
get a<$>(z), ff ( ?W, <r { T(x). To get <r™(x) from
eq 38, in view of eq 40, requires 54— iV multipli-
cations. To get a { %\ <j { \
from eq 40
requires in all approximately (22—4) (30— iV) multi-
plications, where 30 is a rough average of the
higher values of N 2r in eq 40. To put Sg together
by use of eq 34 involves R multiplications.
Summarizing, we find that getting Si by the
outlined procedure requires for each value of z
and each value of x the number of multiplications
W 1 (N,R) = -m + 7N+3lR-RN=
151- (7-22) (31 -N).
(41)
Minimizing Wi(N, R) over the pairs given in eq 32
selects the pair N= 18, 22=5, for which Wi(18, 5) =
125. Assuming that the computation of S 2 in
13 This method is expecially useful with respect to computations on Inter-
national-Business-Machines equipment.
96
Journal of Research
eq 29 involves the same considerations, we may
therefore propose N=IS, R = 5 as being the
optimal values to use in getting G(x, z) by eq 33
and 34 for one value of x and one value of z.
The number of multiplications will be 2W X (N, R).
Getting G(x, z) for 72 values of x and one value
of z involves no change of N, R. Since sin nx
[cos nx] is an odd [even] function of x, the number
of multiplications in getting G(x, z) for all x will
be 72W\(N, R). However, getting G(x, z) for
several values of z changes the analysis, because
the functions o- ( £? (x), once computed, serve for each
new z without change. To get Si for 13 values
of z and one value of x, for example, will require,
in addition to the multiplications in eq 41, only
12iV multiplications from eq 33 and 12R from
eq 34. The total number of multiplications will
then be
W 1Z (N, R) = -66 + l9N+4'3R-NR=
751-(19-i2)(43-2V).
The minimum of Wiz(N, R) is 361, and occurs
for iV=13, R=Q. The optimal choice of N, R
has changed, though not greatly. Since we
expect to use 13 values of z, we adopt the values
JV=13, R--6.
4. Summary of the Computation Method
With the above choice of N and R, the compu-
tation of Si may proceed as follows:
(a) Compute
sin nx.
si =§6 cos C) 9i
(b) For r=0, 1, 2, compute
sin nx
n2r+l >
where <T2r(x) is computed from section VI.
(c) Compute
*£ 3) (x)=-f: s
(42)
~4 n'
^ 13) (*)=S
sin nx
i4 n v
« = 14 n
(43)
(d) Compute
5 z 2r
^2 — 2-1 (<) T \ | °2r \ X J'
r = () K" 1 ) '
(e) Compute 2i=2){+2j.
The number of multiplications involved in get-
ting Si for one x-value and for one .:- value is:
(a) 13; (b) 39; (c) 69; (d) 6; (e) 0. When getting
Si for one x-value and 13 2-values, one adds 156
multiplications to (a) and 72 to (d). The total
for 13 2-values is 355 multiplications per x-value.
(The slight discrepancy with the number 361 in
subsection 3 is due to the rough estimate previously
made for step (c).) For all ^-values (essentially
36), one gets a total of 12,780 multiplications to
get Si.
To get S 2 in eq 29, one follows analogous steps
involving o- 2r+ i (x), <r£!%i(x) t etc. There will be
approximately 12,750 more multiplications, mak-
ing a total of about 25,500 multiplications to get
G(x, z) for the 72 x-values and 11 2-values.
The total truncation error in getting Sj is bounded
by 2X10 -4 . This is divided into four trunca-
tion errors of 5X10 -5 , one for each of the three
steps in (c), and one for the terms left out of (e).
The truncation error for S 2 is also bounded by
2 X 10" 4 , making a total truncation error of 4 X 10~ 4 .
The final round-off of the final answer to three
decimal places may introduce an error of 5X10 -4 .
The third source of error is the accumulation of
round-off s from adding five-decimal-place terms.
Each term is accurate to 5 X 10" 6 ; with an assumed
rectangular distribution these terms have a dis-
persion near 3X10" 6 . Each value of G(x, z) is ob-
tained from the addition of about 270 such terms.
The dispersion a of the sum is therefore about
V270X5X10" 6 , or about 8X10" 5 . One may ex-
pect the accumulated error to exceed 2.5 a=2X
10~ 4 in only 1.3 percent of the cases. The sum
of the three errors is effectively bounded by
11X10" 4 , or slightly more than 0.001.
VI. The Polynomials {^(x)}
In section V we made use of certain functions
a k (x) defined as follows:
0"2
r(*) = (-l) r S
sin nx
r~ »"
(r=0, 1,2,...);
«T 2r+I (x) = (-l)'+ 1 S
cos nx
(r=0,l, 2, ...),
(44)
Solution of the Telegrapher's Equation
97
The function <r (x) was used in section IV; see eq.
18. For &>0 the series a k (x) in eq. 44 are abso-
lutely convergent; hence they represent contin-
uous functions. Since a 2r (x) is odd and (r 2r+1 (x) is
even, it is necessary to sum the series (eq. 44)
only for 0<£<7r.
As stated in eq. 18,
7T 1
<3ro(z) = 2 — 2 X ' (0<^<^)- (45)
No\
; cos nx
(7, (X) = — 2-1 o~
71=1 n 2
Jo w=ir^
Hence
(Ti(x) = — -w--{-yX — jX 2 > (0<X<7r).
Similarly, for 0<^c<x, one finds
1 ,.
t \ W I T 2
<T 2 (X) = —^ X + -^X Z -
12
cr 2\ X ) = CM\~~To X "1
90 12
12
x° —
48
or 4 (a;) =
90
36
a X 3 +
48
240
x°-
(46)
Use has been made of the formulas 2T =1 71 2 =
7r76;S:=i^- 4 = 7r 4 /90.
The functions <r k (x) are therefore all poly-
nomials. Their use in improving the convergence
of Fourier series is pointed out on pp. 84 to 88
of [10]. Although they may be easily tabulated
from eq 46, they may also be adapted from exist-
ing tables because they are essentially Bernoulli
polynomials. Let {B n (x)\ be the Bernoulli poly-
nomials given on p. 181 of [4].
Lemma. 14 For 0^x<7r, andk = Q, 1, 2, . . . ,
(2tt)* +1
(0
**(*)— 2(Jfc+l)! B * +1
Proof: Define the Bernoulli number B n by the
relation B n — B n (0) . These are the Bernoulli num-
bers used on p. 21 of [8]; they are B =l, Bi=— J,
5 2 =l/6,J5 3 -0,5 4 =-l/30,5 5 =0,5 6 -l/42,
Davis uses other notations in [4]. Now fix x in
the interval 0<x<ir. For each k=l, 2, 3, . . . ,
**(*)=* \ <r*-i(&di-"
. Jct ^\ 1
sin — 21/
2 ZAn** 1 '
"After this paper was completed, Professor D. H. Lehmer called the
author's attention to the statement of this lemma on p. 65 of N. E. Norlund's
Vorlesungen fiber Differenzenrechnung (Julius Springer, Berlin, 1924).
Jo '*- i(edf ~2(f+rn
The last step is by eq 9 on p. 21 of [8], which is
correct except for sign. Hence, letting x=2irt,
2(27r)- k - 1 a k (27rt):
s:
B,
2(2ir)-* ff ,_ 1 (2«i|)Ai-^^ I (*=l,2 > 3, . . . ).
(47)
Now define 2(2ir) (, ff_,(27rf) = -B /0\(=-l). Use
eq 47 formally to get 2(2r)~ 1 a (2irt):
2(2x)- 1 <to(2^) = -§^-^ 1 (=-<+|)- (48)
Note that eq 48 agrees with eq 45 for a Q (x). Hence
eq 48 is a correct formula, although it was only
derived formally.
We now apply formula (eq 47) repeatedly,
getting always correct expressions:
/ n_ 2 /o ,\ B t B\ t B 2 .
2{Ztt) < T iV* t )--w.2\~V.V.~W
Of 9 \-3 /o A— ^° ^ Bl ^ ^2 t B 3 .
Z(2ir) a 2 (M)- - Q -y ^-yy 2!~" 2! f!~ 3l''
Hence
w 1 ^ 2 ^— SflFT^r
2(t + 1) l(2,r)-'-V2.f) = -g ( ^f^7
* fk+V
-sCi 1 )^ 1 "-
(49)
But it follows from the top of p. 188 of [4] that
B k+l (t) =S C^r 1 ) Bf-»-i- (50)
Comparing eq 49 and 50, we see that
B k+1 (t) = -2(k + l)\(2ir)- k - 1 a k (27rt).
Let 2irt=x, and the lemma is proved.
VII. Solution by a Difference Equation
Our first approximate solution of the problem
stated in section II consisted of the approximate
evaluation of the integral (eq 21) by means of
numerical integration formulas, using the approxi-
98
Journal of Research
mate values of 0(x, z) tabulated in section VIII
below. A second approximate solution of the
problem consists in solving with appropriate
boundary conditions a difference equation that is
closely related to the differential equation (eq 3).
The latter method is considered in detail in [6],
where proofs may be found; only a summary is
given in the present section.
For any positive integer 2N, let h=w/2N; let
&>0 be arbitrary. A net is formed of all points
(Xj t) of form (fih, vk) , where /x and v are integers
satisfying the conditions
fx+v = (mod 2), \n\<2N, ^>0. (51)
Where necessary we extend the net and the func-
tional values periodically in x with period 2w.
The differential equation (eq 3) is approximated
by the difference equation,
v(x-\-h, t+k)—v(x—h, t-\-k) =
v(x+h, t—k)—v(x—h, t-k)—hkv(x, t). (52)
The boundary conditions of the difference-equa-
tion problem are prescribed values of v(x, t) on the
two rows £=0, t=k. Assume that for t=k,
2>CM)=o,
(53)
where the sum is extended over all points of the
second row of the net. The boundary conditions
and eq 52 then determine the value of v(x, t) on the
row t=2k up to an additive constant. The
additive constant and hence v(x, 2k) are determined
uniquely by requiring that eq 53 hold also for
t = 2k. Continuing row after row, one thus de-
termines v(x, t) over the whole net. Let the
function so determined be denoted by v m (x, f); it
depends on N, on k, and on the initial values pre-
scribed for the first two rows. The problem of [6]
is to see whether v iN) (x, t)-^>v(x, t) as iV— > oo .
Let the initial values v(x, 0) on the first row be
defined by the relation v(x, 0)=f(x), where j(x) is
the function of eq 1. Let k be fixed. Then it is
possible to choose the initial values v(x y k) on the
second row of the net in such a manner that, as
Af-^oo } v m (x, t)->v(x,t) for each t of the net and
for each x that is an abscissa of continuity oij(x).
If k is allowed to vary with N in such a manner
that k—>0 as N-^co ) then v {N) (x, t)-^v{x, t) for each
t>0 and for each x that is an abscissa of conti-
nuity o{J(x) . In neither case may one, in general,
expect the convergence to be uniform in x or t.
The method referred to for choosing the values
v(x, k) on the second row is not an economical
one, and in a practical computation one would
prefer a cheaper though approximate method.
Two things are shown in [6] about the effects of an
approximation of the values of v(x, k) : First, they
may introduce ultimate instability into the solu-
tion. Even though the solution v(x, t) of eq 3
be identically zero, it is possible that for fixed
N and x,
\imv {N) (x,t) = + oo.
Second, the approximation docs not prevent con-
vergence of v {N) (x,t) to v(x,t), provided that the
error of the approximation of v(x,k) vanishes as
iV-^oo. One reasonable way of causing the error
to vanish is to let &— >0.
These results show that the difference-equation
method is a feasible method of solving the
problem of this paper.
VIII. Table of the Green's Function
In this section is tabulated the Green's function
G(x, z), as computed in the Computation Unit of
the Institute for Numerical Analysis. The value of
the time parameter z corresponding to h hours at
latitude <j> is
2=0.52502 h cos 2 <j>. (51)
(Except for the last digit of the constant, formula
(eq 51) can be verified from the introduction to
section V.) Meteorological considerations sug-
gested that h should be chosen in convenient mul-
tiples of 12 hours, and that </> should be 35°, 45°, or
55°. The latitude 32°19' resulted from a numer-
ical error by the author. A limited number of
pairs of values of h and <j> were selected for the
computation; these pairs arc shown in table 1,
together with the corresponding values of z
determined from eq 51.
For each of the 13 values of z given in table 1
(and for z=0) and for x=— x(7r/36)ir, the Green's
function G(x, z) is presented in table 2 to 5 deci-
mal places. Since 6? (#, z) has a discontinuity of
the first kind at x=0, the values c7( — 0, z) } G(0, z),
and 67(+0, z) arc all given. In every case,
67(0, 3)=i[G(-0, z) + G(0, z)] and G(+0, z) —
G( — 0, z) = 7r. The computational procedure fol-
lowed that of section V in general outline, with
certain deviations. It was decided to use N=18,
i?=6. The auxiliary functions o- ( £? (x) and
Solution of the Telegrapher's Equation
99
o"2r+\ 0*0 were computed from formulas like eq 36;
formulas like eq 38 were not used. This necessi-
tated carrying considerably more than 10 digits,
and so the polynomials <r k (x) were first computed
to 17 decimal places. Choice of formula (eq 36)
was based on the value of getting these tables of
Bernoulli polynomials as a byproduct of general
interest.
To be definite/ 5 let us write G{x, z) = G 1 (x, 2) +
G 2 (x, z) , where
G 1 (x i z)=T,lsm(nx+^\ (52)
and
G 2 (x,z) = ^2 -sin(nx-\ — )•
Let
(b k /dz k )G 2 (x,z) = G ifc) (x,z),tork=0, 1,2, ....
Then
G {k) (x, 0)=S -tt sin (nx+Ic %)=g k (x) —h k (x),
n =iQ n ^ \ zy
where
0*0*0 =S ^ffi sin (nx+k | j,
By section VI, the functions ^(z) are polynomials.
They were generated on an International-Busi-
ness-Machines tabulator to 17 decimal places.
The values of h k (x) were computed and subtracted
from g k (x) to yield G (k) (x, 0). For any z,
G 2 (x,z)=Jly ] GM(x,0)+R k (x ) z),
where there exists a z x (0<2!<2) such that
z h
(53)
|ff*(*,2)l<-
-,K+1
^ (K+ "(X, g,)-
(K+l)!
If 0<z<36,2 n /ll!<.33X10 10 . And \G iu \x, Si)|<
S«" 12 <1-5X10- 15 . Hence, for 0<z< 36,
|fl„(x,g)l< 0.5X10- 5 .
15 This description of the computational procedure was furnished by
Gertrude Blanch of the Computation Unit, Institute for Numerical Analysis
Once the values of G (k) (x, 0) were obtained it was
possible to generate, very easily, the function
G(x,z) for any values of z in the range 0<2<36.
To summarize,
(a) Gi(x, z) was computed from eq (52).
(b) The functions G ik) (x, 0) were computed
for £==0,1,2, . . . , 10.
(c) G 2 (x } z) was obtained from eq (53).
(d) Gi(x, z) and G 2 (x, z) were added, to yield
G(x, z).
The subsidiary computations in (a) and (b) were
carried to nine decimal places, those in (c) to at
least seven decimal places in the partial products.
Table 2 is believed to be accurate to ±0.00002
for all x and all z. The cosine component of
G(x, z) was given a final check by use of the fol-
lowing formula:
35 oo n -
S G{k*lW J z)=^±-sm~= V {z)-
fe = -36 v = l v
72/
The check showed a deviation between the sum
and p(z), which was never greater than 0.00005.
The sine component of G (x, z) was given a partial
check by the formula
35 k
XI sin -pr- G(Jct/36,z) =2 cos 7-5+
k = -3Q Z 15
2±T- 1
L>+i
cos
(:
8/18 \ 1_
4v+l) 4?— 1
cos
am
Comparable agreement was found. Finally, the
table differences with respect to x are very
satisfactory.
Table 1. Values of the time parameter, z, corresponding
to hours, h, at latitude, <f>
4>
h
32°19'
35°
45°
55°
Hours
12
3. 150
24
9.000
8.455
6.300
4.145
48
18. 000
16. 909
12.600
8.291
72
27. 000
18. 900
96
36. 000
25. 200
100
Journal of Research
Table 2. Table of Green's Function G(x, z).
Table 2. Table of Green's Function G(x, z).
2
X
0.00000
3.15012
4.14543
6.30024
8.29086
8.45505
9.00000
Degrees
-180
-0.00000
0. 32695
1.08321
-0. 16762
-1.34038
-1.26672
-0. 85722
-175
-. 04363
. 39530
1.08444
-.30315
-1.29132
-1.19935
-.74063
-170
-.08727
. 45756
1.07647
-. 43082
-1. 22533
-1.11601
-.61286
-165
-. 13090
. 51370
1.05993
—.54955
-1.14399
-1.01848
-.47630
-1G0
-. 17453
. 56372
1.03552
-.65844
-1.04899
-0. 90869
-.33337
-155
-.21817
. 60768
1.00392
-.75679
-0. 94217
-.78862
-.18646
-150
-.26180
. 64565
0. 96581
-. 84405
-.82542
-. 66032
-.03791
-145
-.30543
. 67775
. 92189
-. 91985
-. 70067
-. 52583
. 11005
-140
-. 34907
. 70413
. 87284
-.98398
-. 56985
-.38717
. 25533
-135
-. 39270
. 72496
. 81933
-1.03636
-. 43487
-. 24630
. 39595
-130
-. 43633
. 74042
. 76202
-1.07704
-. 29757
-. 10512
. 53015
-125
-. 47997
. 75072
. 70154
-1.10620
-. 15975
. 03460
. 65632
-120
-. 52360
. 75611
. 63851
-1.12412
-.02308
.17118
. 77306
-115
-. 56723
. 75679
. 57351
-1.13116
. 11085
. 30309
. 87916
-110
-. 61087
. 75304
. 50712
-1.12779
. 24058
. 42893
. 97363
-105
-. 65450
. 74510
. 43985
-1.11454
. 36479
. 54746
1. 05568
-100
-. 69813
. 73324
. 37221
-1.09199
. 48229
. 65759
1. 12472
-95
-. 74176
.71771
. 30468
-1.06078
. 59203
. 75838
1. 18034
-90
-. 78540
. 69880
. 23769
-1.02 ICO
. 69310
. 84906
1. 22233
-85
-. 82903
. 67677
. 17165
-0.97516
. 78475
. 92902
1. 25066
-80
-. 87266
. 65189
. 10695
-.92219
. 86637
. 99779
1.26545
-75
-. 91630
. 62442
. 04392
-.86344
. 93749
1. 05505
1. 26700
-70
-0. 95993
. 59462
-.01712
-.79967
. 99777
1. 10063
1. 25573
-65
-1.O0356
. 56277
-.07590
-.73162
1. 04701
1. 13450
1.23218
-60
-1.04720
. 52910
-.13215
-.66004
1. 08513
1.15674
1.19701
-55
-1.09083
. 49386
-.18566
-.58566
1.11218
1. 16755
1. 15098
-50
-1.13446
. 45730
-.23624
-. 50920
1.12832
1. 16724
1.09494
-45
-1.17810
. 41965
-.28373
-.43136
1. 13378
1.15623
1. 02978
-40
-1.22173
.38113
-.32798
-.35278
1. 12892
1. 13502
0. 95646
-35
-1.26536
. 34197
-.36889
-.27411
1. 11416
1. 10417
. 87600
-30
-1.30900
. 30236
-.40638
-.19596
1. 09002
1.06431
. 78941
-25
-1.35263
. 26250
-.44037
-. 11887
1. 05705
1. 01616
. 69774
-20
-1.39626
. 22259
-.47084
-.04338
1. 01588
0. 96044
. 60203
-15
-1.43990
. 18280
-.49776
.03002
0. 96717
. 89794
. 50333
-10
-1.48353
. 14330
-.52114
. 10089
.91164
. 82944
.40264
-5
-1.52716
. 10426
-.54100
. 16881
. 85000
. 75578
.30096
-0
-1.57080
.06582
-.55736
. 23344
. 78302
. 67778
. 19926
0. 00000
1. 63662
1. 01344
1.80424
2. 35382
2. 24858
1. 77006
+0
1. 57080
3. 20741
2. 58423
3. 37503
3. 92461
3. 81938
3. 34085
5
1. 52716
2. 36366
1. 53355
1.93219
1. 95959
1. 81396
1. 21684
10
1. 48353
1. 62904
0. 66827
0. 87990
0. 63440
0. 47195
-0. 16296
15
1.43990
0. 99385
-.03330
. 14729
-. 20075
-. 36465
-. 98574
20
1. 39626
. 44908
-.59093
-.32689
-. 66889
-. 82505
-1.40257
25
1. 35263
-.01369
-1.02264
-. 59533
-. 87018
-1. 01399
-1. 53397
30
1.30900
-. 40226
-1.34473
-. 70310
-. 88521
-1.01530
-1. 47484
35
1. 26536
-. 72390
-1.57202
-.68848
-. 77804
-0. 89523
-1. 29869
40
1. 22173
-.98534
-1.71788
-. 58382
-. 59873
-. 70520
-1. 06130
45
1. 17810
-1.19282
-1.79437
-.41615
-. 38563
-. 48429
-0. 80387
50
1. 13446
-1.35212
-1.81235
-.20791
-. 16734
-. 26137
-. 55574
55
1.09083
-1.46858
-1.78153
. 02253
. 03555
-.05690
-.33671
60
1.04720
-1.54714
-1.71063
. 26034
. 20903
. 11545
-.15898
65
1. 00356
-1.59233
-1.60740
. 49376
. 34437
. 24746
-.02884
70
0. 95993
-1.60833
-1.47871
.71367
. 43706
. 33521
. 05197
75
. 91630
-1.59898
-1.33066
.91321
. 48585
. 37809
. 08507
80
.87266
-1.56780
-1.16858
1. 08748
. 49200
. 37799
. 07448
z
X
0. 00000
3. 15012
4. 14543
6. 30024
8. 29086
8. 45505
9.00000
Degrees
85
. 82903
-1.51800
-0. 99717
1. 23323
. 45862
.33860
.02585
90
. 78540
-1.45251
-. 82049
1.34855
.39011
. 26486
-.05417
95
.74176
-1.37403
-. 64207
1.43272
. 29171
.16247
-. 15846
100
. 69813
-1.28496
-. 46492
1.48592
. 16915
.03753
-.27984
105
. 65450
-1.18751
-.29158
1. 50910
. 02832
-. 10376
-.41132
110
. 61087
-1.08367
-. 12420
1. 50380
-. 12493
-. 25537
-. 54637
115
. 56723
-0. 97523
. 03545
1. 47204
-. 28498
-.41155
-.67907
120
. 52360
-.86377
. 18594
1.41614
-. 44662
-. 56703
-.80424
125
. 47997
-.75074
.32616
1.33870
-.60507
-. 71708
-.91745
130
. 43633
-.63740
. 45522
1. 24244
-.75611
-. 85757
-1.01507
135
. 39270
-.52487
. 57252
1. 13016
-.89609
-.98499
-1.09431
140
. 34907
-.41413
. 67764
1.00467
-1.02197
-1.09649
-1.15312
145
. 30543
-.30603
. 77037
0. 86874
-1.13129
-1.18985
-1.19019
150
. 26180
-.20133
. 85064
. 72505
-1.22217
-1.26346
-1.20491
155
.21817
-.10064
. 91855
. 57617
-1.29329
-1.31627
-1.19725
100
. 17453
-.00450
. 97429
. 42452
-1.34386
-1.34779
-1.16777
165
. 13090
. 08665
1.01819
. 27234
-1.37355
-1.35800
-1.11746
170
.08727
. 17245
1. 05065
.12170
-1.38248
-1.34730
-1.04773
175
. 04363
. 25262
1.07214
-.02551
-1.37114
-1.31650
-0. 96032
180
. 00000
. 32695
1. 08321
-. 16762
-1.34038
-1.26672
-.85722
12.60048
16.91010
18.00000
18.90072
25.20096
27.00000
36.00000
-180
0. 21759
1. 38333
0.88710
-0. 22251
-0. 23708
-0. 52214
0. 75323
-175
. 20848
1.28719
. 68046
-.44845
-. 13949
-.34513
. 66966
-170
.18520
1.17912
.47714
-. 65407
-.04531
-.19161
. 59996
-165
.14768
1.06395
. 28213
-. 83538
. 03742
-.06711
. 55216
-160
.09639
0.94615
. 09946
-.98956
. 10193
.02510
. 53133
-155
. 03224
. 82961
-.0674';
-1.11498
. 14294
.08392
. 53944
-150
-.04345
.71765
-.21599
-1. 21108
. 15684
.11027
. 57545
-145
-.12901
. 61292
-. 34473
-1.27829
.14173
. 10689
. 63557
-140
-. 22253
. 51745
-. 45323
-1.31787
. 09736
. 07795
. 71381
-135
-.32192
. 43256
-.54152
-1.33178
. 02503
. 02877
. 80257
-130
-.42494
. 35899
-.61014
-1.32256
-.07262
-. 03461
. 89326
-125
-. 52932
. 29691
-. 66035
-1.29314
-. 19180
-.10578
. 97703
-120
-. 63276
. 24596
-.69412
-1.24669
-. 32784
-. 17834
1. 04492
-115
-.73306
. 20536
-. 71352
-1.18651
-.47538
-. 24602
1.09064
-110
-.82808
. 17397
-.72062
-1.11592
-. 62883
-.30340
1. 10661
-105
-.91587
. 15038
-.71767
-1.03810
-.78241
-.34564
1. 08871
-100
-. 99463
. 13297
-. 70728
-0. 95602
-. 93051
-. 36894
1. 03429
-95
-1.06278
. 12001
-.69185
-. 87240
-1.06785
-.37053
0. 94265
-90
-1. 11899
. 10975
-. 67337
-. 78959
-1.18968
-.34876
.81504
-85
-1. 16216
. 10043
-. 65362
-. 70959
-1.29191
-. 30309
.65453
-80
-1. 19145
. 09041
-. 63445
-.63398
-1.37124
-.23406
. 46575
-75
-1.20628
. 07817
-. 61745
-. 56396
-1.42518
-. 14319
.25459
-70
-1.20633
. 06238
-. 60358
-. 50029
-1.45213
-.03286
. 02788
-65
-1.19154
. 04192
-. 59343
-. 44339
-1.45134
.09382
-. 20697
-60
-1. 16210
. 01592
-. 58753
-. 39331
-1.42289
.23321
-. 44192
-55
-1. 11840
-.01623
-. 58625
-.34978
-1.36773
.38109
-.67081
-50
-1.06108
-. 05488
-. 58940
-.31226
-1.28738
.53326
-.88534
-45
-0. 99094
-.10011
-. 59636
-. 28000
-1.18407
. 68527
-1.07970
-40
-.90897
- 15176
-. 60651
-.25205
-1.06049
. 83281
-1.24859
-35
-.81632
-. 20939
-. 61926
-.22736
-0.91976
. 97176
-1.38784
Solution of the Telegrapher's Equation
101
Table 2. Table of Green's Function G(x, z).
2
X
12.60048
16.91010
18.00000
18.90072
25.20096
27.00000
36.00000
Degrees
-30
-. 71423
-.27239
- 63367
- 20477
-. 76527
1. 09836
-1. 49453
-25
-.60408
-. 33990
-.64843
-.18315
-.60056
1. 20927
-1. 56697
-20
-. 48731
-. 41091
-.66232
-. 16133
-.42926
1. 30167
-1. 60479
-15
-.36540
-. 48428
-.67437
-. 13823
-.25493
1. 37335
-1. 60877
-10
-. 23989
-. 55874
-.68350
-.112S6
-. 08102
1. 42266
-1. 58O80
-5
-.11230
-.63295
-.68839
-.08436
. 08924
1. 44862
-1. 52370
-0
.01585
-. 70554
-.68784
-.05201
. 25293
1. 45084
-1.44108
1. 58665
. 86526
. 88296
1. 51879
1.82373
3. 02164
0. 12972
+0
3. 15744
2. 43605
2. 45375
3. 08958
3. 39452
4. 59243
1. 70O52
5
0. 67142
-0. 81525
-0. 84342
-0. 27249
-0. 37260
0. 54092
-2. 54939
10
-. 51209
-2. 00417
-1.87971
-1. 21362
-.63490
. 27468
-1. 76069
15
-. 85078
-2. 07925
-1. 75697
-0. 97035
. 26636
1.10072
-0. 49519
20
-. 68020
-1.65134
-1. 16076
-. 29000
1. 20809
1.82328
-. 00356
25
-.24015
-1. 09098
-0. 48829
. 4121.3
1. 76793
2. 06387
-. 20922
30
. 30379
-0. 60220
. 05701
. 93791
1.89206
1. 84177
-. 68549
35
. 84267
-.27771
. 39671
1. 22420
1.69102
1. 32871
-1.06815
40
1. 31002
-.13946
. 52770
1.28534
1.32178
0. 71692
-1. 17421
45
1. 67048
-.16770
. 48736
1. 17420
0. 92635
. 15777
-0. 98652
50
1. 91086
-.32146
. 33030
0. 95687
. 60563
-. 25779
-. 59209
55
2. 03312
-. 55228
. 11399
.69716
.41414
-. 49626
-. 11820
60
2. 04891
-.81299
-.11013
. 44820
. 36577
-. 56829
. 31487
65
1. 97550
-1. 06293
-.30155
. 24868
. 44447
-. 51226
. 62341
70
1. 83275
-1.27046
-. 43250
. 12229
. 61561
-. 37956
. 76541
75
1. 64092
-1. 41383
-.48723
. 07901
. 83661
-. 22221
. 74192
80
1.41914
-1. 48076
-.46049
. 11749
1. 06504
-. 08486
. 58453
85
1.18449
-1. 46731
-.35588
. 22780
1. 26435
. 00000
. 34287
90
0. 95137
-1.37647
-. 18386
. 39425
1. 40727
. 01370
. 07206
95
. 73134
-1.21644
. 04076
. 59799
1.47711
-.04915
-. 17737
100
. 53301
-0. 99910
. 30108
.81915
1. 46763
-. 18262
-.36632
105
. 36223
-. 73849
. 57953
1. 03865
1. 38174
-. 37234
-.47053
110
. 22235
-.44954
. 85891
1. 23947
1. 22959
-. 59864
-.48116
115
. 11450
-. 14704
1.12358
1. 40751
1. 02640
-.83951
-. 40310
120
. 03798
. 15526
1.36058
1. 53214
0. 79018
-1.07324
-.25141
125
-.00939
. 44516
1. 55988
1. 60630
. 53965
-1.28065
-. 04972
130
-. 03091
. 71247
1.71412
1. 62645
. 29276
-1. 44613
. 17798
135
-. 03066
. 94918
1.81868
1. 59230
. 06507
-1.55895
. 40643
140
-.01322
1. 14954
1.87196
1 . 50636
-. 13095
-1. 61332
. 61405
145
. 01665
1.31003
1.87498
1. 37347
-.28656
-1. 60827
. 78435
150
. 05419
1.42912
1.83068
1.20024
-.39683
-1. 54718
. 90692
155
. 09488
1. 50714
1.74337
0. 99449
-. 46051
-1. 43694
. 97765
160
. 13453
1. 54593
1.61879
. 76479
-.47971
-1.28710
. 99822
165
. 16945
1. 54859
1 . 46378
. 51991
-.45930
-1. 10889
. 97514
170
. 19650
1. 51918
1 . 28553
. 26848
-.40628
-0. 91426
. 91844
175
.21318
1. 46239
1.09103
. 01857
-.32913
-. 71502
. 84021
180
.21759
1.38333
0. 88710
-.22251
-.23708
-. 52214
. 75323
IX. References
[I] J. Charney, A. Eliassen, G. Hunt, Report on a program
for numerical weather predicting, paper presented
at a meeting of the American Meteorological Society,
New York, N. Y., January 28, 1949.
[2] J. Charney, manuscript in preparation for Journal
of Meteorology on the work reported in [1].
[3] R. Courant, Differential and integral calculus (trans-
lated by E. J. McShane) 2 (Blackie and Sons,
London and Glasgow, 1936).
[4] Harold T. Davis, Tables of the higher mathematical
functions, 2 (Principia Press, Bloomington, Ind.,
1935).
[5] G. E. Forsythe, J. Meteorology 4, 67 (1947).
[6] G. E. Forsythe, Manuscript in preparation on the
approximate solution by difference equations of the
telegrapher's equation with boundary values on only
one characteristic.
[7] G. E. Forsythe, A solution of the telegrapher's equation
with initial conditions on only one characteristic.
Preliminary Report. Paper presented at a meeting
of American Mathematical Society, Columbus, Ohio,
December 30, 1948.
[8] Erwin Madelung, Die Mathematischen Hilfsmittel des
Physikers, American ed. (Dover Publications, New
York, N. Y., 1943).
[9] C. G. Rossby and collaborators, J. Marine Res. 2, 38
(1939).
[10] J. D. Tamarkin and Willy Feller, Partial differential
equations (mimeographed lecture notes, Brown
University, Providence, R. I., 1941).
[II] A. G. Webster, Partial differential equations of
mathematical physics, 2d corrected ed. (Hafner
Publishing Co., New York, N. Y., 1947).
[12] A. Zygmund, Trigonometrical Series (Monografje
Matematyczne, Warsaw-Lwow, 1935).
Los Angeles, March 8, 1949.
102
Journal of Research